Physics 4820: Mathematical Physics II
4820 Mathematical Physics II covers topics on partial differential equations of Mathematical Physics and boundary value problems; Sturm-Liouville theory, introduction to the theory of distributions, Dirac delta function, Laplace and Fourier transforms, Green’s functions, Bessel functions, Legendre functions, spherical harmonics, and other topics such as group theory.
PR: Physics 3820 or all of Mathematics 2051, 2260, 3202, 3210
Keep calm and solve differential equations (http://sd.keepcalm-o-matic.co.uk/).
This course uses the mathematical tools and methods taught in Physics 3820. In many cases, in physics, solving problems involves formulating them in terms of differential equations in one-, two-, three- or more dimensions. Examples of important differential equations are: Schrodinger equation in quantum mechanics, Laplace and Poisson equations in electricity and magnetism, and acoustics, diffusion equation in fluid mechanics, wave equation in theory of elasticity and optics, Dirac’s relativistic field theory equations, gauge theory equations and many others. As stated these equations are very difficult to solve. In Physics 4820 students will be introduced to multiple techniques that will allow them to “solve” these equations and at the same time gain insight into the nature of the solutions. We’ll find that the separation of variables technique is a powerful tool to reduce three-dimensional problem to one-dimensional ODEs. We’ll learn that orthogonal functions such as Bessel functions and Legendre polynomials allows us to formulate solutions to difficult problems in a consistent and accessible way. The use of integral transforms often reduces a very difficult to “easy” problem. For example, we will see that solving problems such as damped harmonic oscillator or driven electrical circuit can be readily carried out with the use of Laplace transforms. Symmetry plays a very big role in many branches of physics (e.g. quantum mechanics, elementary particle physics, crystallography and many others). Symmetry in physics is often expressed with the help of group theory. That is the reason for introducing it in this course. The use of symmetry greatly simplifies complex problems such as, for example, determining the crystal structures and makes solving the appropriate differential equations tractable. There are two preferred (complementary) textbooks for this course: Mathematics for Physicists, S. M. Lea (Thomson, 2004) and Mathematical Methods for Physicists A Comprehensive Guide, Afken, Weber and Harris (Elsevier Academic Press, 2013).